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Theorem archiabl 30082
 Description: Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
Assertion
Ref Expression
archiabl ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Proof of Theorem archiabl
Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2760 . . . . 5 (0g𝑊) = (0g𝑊)
3 eqid 2760 . . . . 5 (le‘𝑊) = (le‘𝑊)
4 eqid 2760 . . . . 5 (lt‘𝑊) = (lt‘𝑊)
5 eqid 2760 . . . . 5 (.g𝑊) = (.g𝑊)
6 simpll1 1255 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
7 simpll3 1259 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
8 simplr 809 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑣 ∈ (Base‘𝑊))
9 simprl 811 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → (0g𝑊)(lt‘𝑊)𝑣)
10 simp2 1132 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑦 ∈ (Base‘𝑊))
11 simp1rr 1306 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
12 simp3 1133 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → (0g𝑊)(lt‘𝑊)𝑦)
13 breq2 4808 . . . . . . . 8 (𝑥 = 𝑦 → ((0g𝑊)(lt‘𝑊)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑦))
14 breq2 4808 . . . . . . . 8 (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥𝑣(le‘𝑊)𝑦))
1513, 14imbi12d 333 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1615rspcv 3445 . . . . . 6 (𝑦 ∈ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) → ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1710, 11, 12, 16syl3c 66 . . . . 5 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑣(le‘𝑊)𝑦)
181, 2, 3, 4, 5, 6, 7, 8, 9, 17archiabllem1 30077 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
1918adantllr 757 . . 3 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
20 simpr 479 . . . 4 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
21 breq2 4808 . . . . . 6 (𝑢 = 𝑣 → ((0g𝑊)(lt‘𝑊)𝑢 ↔ (0g𝑊)(lt‘𝑊)𝑣))
22 breq1 4807 . . . . . . . 8 (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
2322imbi2d 329 . . . . . . 7 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2423ralbidv 3124 . . . . . 6 (𝑢 = 𝑣 → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2521, 24anbi12d 749 . . . . 5 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))))
2625cbvrexv 3311 . . . 4 (∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2720, 26sylib 208 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2819, 27r19.29a 3216 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
29 simpl1 1228 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
30 simpl3 1232 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
31 eqid 2760 . . 3 (+g𝑊) = (+g𝑊)
32 simpl2 1230 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → (oppg𝑊) ∈ oGrp)
33 simpr 479 . . . . . . . . . 10 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
34 ralnex 3130 . . . . . . . . . 10 (∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3533, 34sylibr 224 . . . . . . . . 9 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
36 rexanali 3136 . . . . . . . . . . . 12 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))
3736imbi2i 325 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
38 imnan 437 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3937, 38bitri 264 . . . . . . . . . 10 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4039ralbii 3118 . . . . . . . . 9 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4135, 40sylibr 224 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)))
4222notbid 307 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
4342anbi2d 742 . . . . . . . . . . 11 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4443rexbidv 3190 . . . . . . . . . 10 (𝑢 = 𝑣 → (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4521, 44imbi12d 333 . . . . . . . . 9 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥))))
4645cbvralv 3310 . . . . . . . 8 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4741, 46sylib 208 . . . . . . 7 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4847r19.21bi 3070 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4914notbid 307 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
5013, 49anbi12d 749 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
5150cbvrexv 3311 . . . . . 6 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
5248, 51syl6ib 241 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
53523impia 1110 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
54 simp1l1 1351 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ oGrp)
55 isogrp 30032 . . . . . . 7 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
5655simprbi 483 . . . . . 6 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57 omndtos 30035 . . . . . 6 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
5854, 56, 573syl 18 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ Toset)
59 simp2 1132 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑣 ∈ (Base‘𝑊))
601, 3, 4tltnle 29992 . . . . . . . . . 10 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
6160bicomd 213 . . . . . . . . 9 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
62613com23 1121 . . . . . . . 8 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
63623expa 1112 . . . . . . 7 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
6463anbi2d 742 . . . . . 6 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6564rexbidva 3187 . . . . 5 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6658, 59, 65syl2anc 696 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6753, 66mpbid 222 . . 3 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
681, 2, 3, 4, 5, 29, 30, 31, 32, 67archiabllem2 30081 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
6928, 68pm2.61dan 867 1 ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   class class class wbr 4804  ‘cfv 6049  Basecbs 16079  +gcplusg 16163  lecple 16170  0gc0g 16322  ltcplt 17162  Tosetctos 17254  Grpcgrp 17643  .gcmg 17761  oppgcoppg 17995  Abelcabl 18414  oMndcomnd 30027  oGrpcogrp 30028  Archicarchi 30061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-tpos 7522  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-seq 13016  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-plusg 16176  df-ple 16183  df-0g 16324  df-preset 17149  df-poset 17167  df-plt 17179  df-toset 17255  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-sbg 17648  df-mulg 17762  df-oppg 17996  df-cmn 18415  df-abl 18416  df-omnd 30029  df-ogrp 30030  df-inftm 30062  df-archi 30063 This theorem is referenced by: (None)
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